Geodesic
Inverse spectral reconstruction problem for the convolution operator perturbed by a one-dimensional operator
Matematičeskie zametki, Tome 80 (2006) no. 5, pp. 668-682

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We consider a one-dimensional perturbation of the convolution operator. We study the inverse reconstruction problem for the convolution component using the characteristic numbers under the assumption that the perturbation summand is known a priori. The problem is reduced to the solution of the so-called basic nonlinear integral equation with singularity. We prove the global solvability of this nonlinear equation. On the basis of these results, we prove a uniqueness theorem and obtain necessary and sufficient conditions for the solvability of the inverse problem. 
@article{MZM_2006_80_5_a2,
     author = {S. A. Buterin},
     title = {Inverse spectral reconstruction problem for the convolution operator perturbed by a one-dimensional operator},
     journal = {Matemati\v{c}eskie zametki},
     pages = {668--682},
     publisher = {mathdoc},
     volume = {80},
     number = {5},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_5_a2/}
}
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S. A. Buterin. Inverse spectral reconstruction problem for the convolution operator perturbed by a one-dimensional operator. Matematičeskie zametki, Tome 80 (2006) no. 5, pp. 668-682. http://geodesic.mathdoc.fr/item/MZM_2006_80_5_a2/